Resource effective control of Elymus repens

Preliminary results show that there is room for improvement within existing control methods of couch grass (Elymus repens (L.) Gould). It may be possible to reduce the number of stubble cultivations during autumn by timing the treatment, and to reduce the cultivation depth by using a goose foot cultivator (5 cm) instead of a disc cultivator (10 cm), without sacrificing couch grass control efficiency. The first year of the experiment, the use of a goose foot cultivator resulted in less nitrogen leaching than cultivation by disc. A reduced number of stubble cultivations potentially reduces nutrient loss, fuel consumption and the workload of the farmer. Our experiments with cover crops to control couch grass in cereals has yet to prove significant effects on couch grass control, but cover crops combined with goose foot hoeing did reduce nitrogen leaching by more than a third compared to cultivation by disc. Further data is necessary to see if the system can be used to effectively control couch grass without significant yield losses. Regardless, it can reduce nitrogen leaching and potentially provide other ecosystem services, e.g. control weeds other than couch grass

A Holder Continuous Nowhere Improvable Function with Derivative Singular Distribution

We present a class of functions $\mathcal{K}$ in $C^0(\R)$ which is variant of the Knopp class of nowhere differentiable functions. We derive estimates which establish \mathcal{K} \sub C^{0,\al}(\R) for 0<\al<1 but no $K \in \mathcal{K}$ is pointwise anywhere improvable to C^{0,\be} for any \be>\al. In particular, all $K$'s are nowhere differentiable with derivatives singular distributions. $\mathcal{K}$ furnishes explicit realizations of the functional analytic result of Berezhnoi. Recently, the author and simulteously others laid the foundations of Vector-Valued Calculus of Variations in $L^\infty$ (Katzourakis), of $L^\infty$-Extremal Quasiconformal maps (Capogna and Raich, Katzourakis) and of Optimal Lipschitz Extensions of maps (Sheffield and Smart). The "Euler-Lagrange PDE" of Calculus of Variations in $L^\infty$ is the nonlinear nondivergence form Aronsson PDE with as special case the $\infty$-Laplacian. Using $\mathcal{K}$, we construct singular solutions for these PDEs. In the scalar case, we partially answered the open $C^1$ regularity problem of Viscosity Solutions to Aronsson's PDE (Katzourakis). In the vector case, the solutions can not be rigorously interpreted by existing PDE theories and justify our new theory of Contact solutions for fully nonlinear systems (Katzourakis). Validity of arguments of our new theory and failure of classical approaches both rely on the properties of $\mathcal{K}$.Comment: 5 figures, accepted to SeMA Journal (2012), to appea

Convexity criteria and uniqueness of absolutely minimizing functions

We show that absolutely minimizing functions relative to a convex Hamiltonian $H:\mathbb{R}^n \to \mathbb{R}$ are uniquely determined by their boundary values under minimal assumptions on $H.$ Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in $L^\infty.$Comment: 34 page

Existence, uniqueness and structure of second order absolute minimisers

Let ⊆ Rn be a bounded open C1,1 set. In this paper we prove the existence of a unique second order absolute minimiser u∞ of the functional E∞(u, O) := F(·, u)L∞(O), O ⊆ measurable, with prescribed boundary conditions for u and Du on ∂ and under natural assumptions on F. We also show that u∞ is partially smooth and there exists a harmonic function f∞ ∈ L1() such that F(x, u∞(x)) = e∞ sgn f∞(x) for all x ∈ { f∞ = 0}, where e∞ is the infimum of the global energy

Existence and uniqueness of global solutions to fully nonlinear second order elliptic systems

We consider the problem of existence and uniqueness of strong a.e. solutions u:Rn⟶RNu:Rn⟶RN to the fully nonlinear PDE system F(⋅,D2u)=f, a.e. on Rn,(1) F(⋅,D2u)=f, a.e. on Rn,(1) when f∈L2(Rn)Nf∈L2(Rn)N and F is a Carathéodory map. (1) has not been considered before. The case of bounded domains has been studied by several authors, firstly by Campanato and under Campanato’s ellipticity condition on F. By introducing a new much weaker notion of ellipticity, we prove solvability of (1) in a tailored Sobolev “energy” space and a uniqueness estimate. The proof is based on the solvability of the linearised problem by Fourier transform methods, together with a “perturbation device” which allows to use Campanato’s near operators. We also discuss our hypothesis via counterexamples and give a stability theorem of strong global solutions for systems of the form (1)

A nonhomogeneous boundary value problem in mass transfer theory

We prove a uniqueness result of solutions for a system of PDEs of Monge-Kantorovich type arising in problems of mass transfer theory. The results are obtained under very mild regularity assumptions both on the reference set $\Omega\subset\mathbf{R}^n$, and on the (possibly asymmetric) norm defined in $\Omega$. In the special case when $\Omega$ is endowed with the Euclidean metric, our results provide a complete description of the stationary solutions to the tray table problem in granular matter theory.Comment: 22 pages, 2 figure

An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions

We present a new, easy, and elementary proof of Jensen's Theorem on the uniqueness of infinity harmonic functions. The idea is to pass to a finite difference equation by taking maximums and minimums over small balls.Comment: 4 pages; comments added, proof simplifie